Structural health monitoring systems (SHM) and other damage detection systems use guided wave inspection methods for testing large structures. Guided ultrasonic waves are popular because they travel through the thickness of the structure over long distances with little attenuation. This allows sensors to interrogate large areas all at once. However, guided waves are inherently multi-modal and dispersive in their propagation. Furthermore, structures' boundaries generate reflections and exchange energy between wave modes. These effects make the interpretation of measured data difficult and necessitate the use of baseline measurements with no damage present.
Under ideal conditions, SHM techniques easily detect damage as a change in the propagation medium by performing a subtraction or time domain correlation with a baseline signal. The baseline reduces complexity by removing effects from static sources, such as reflecting boundaries. Unfortunately, when the environmental conditions change, these methods are unable to distinguish damage from benign effects.
Temperature Compensation
One of the most prominent environmental effects to distort signals is temperature fluctuation. Temperature has been shown to modify the velocity of the guided wave modes, which stretches or scales the measured time domain signal. Given an ultrasonic signal measurement x(t), a change in temperature T{⋅} can be modeled as a uniform scaleT{x(t)}=x(αt),  (1)where α is an unknown scaling factor. This effect can be attributed to temperature's influence on the Young's modulus of the material.
However, this model is an approximation. In the presence of multi-modal and dispersive propagation, temperature's effects do not perfectly dilate or compress the signal. Even so, the effectiveness of this approximate model has been experimentally demonstrated in references. This implies that, by computing a scale-invariant measure of similarity between the baseline signal and measured data, one can compensate for temperature's effects on the ultrasonic signals. Two prior art techniques have been proposed for ultrasonic temperature compensation by computing a scale-invariant statistic: (a) optimal signal stretch, and (b) local peak coherence.
Optimal Signal Stretch
Optimal signal stretch (OSS) is prior art technique for ultrasonic temperature compensation that uses an exhaustive search optimization strategy for finding the scale factor α which minimizes the mean squared error between baseline and measured data. OSS employs a finite library of K stretched baselines {s(α1 t), s(α2 t), . . . , s(αK t)} and computes the mean squared error between the observed signal and each element in the library. The scale factor which minimizes the mean squared error is then declared to be the optimal choice over the given set. As K→∞, OSS computes the optimal scale factor, over all possible scale factors, that minimizes the mean squared error.
When the baseline and observed data are normalized to have zero means and equal L2 norms, minimizing the mean squared error is equivalent to maximizing the sample Pearson product-moment correlation coefficient, or correlation coefficient for short. In the use of OSS, data is normalized so that changes in the signal mean or amplitude, which generally do not correspond with damage, do not bias the results. With normalizing the data, the optimal scale factor α chosen by OSS is defined by
                              α          =                      arg            ⁢                                          max                k                            ⁢                                                ∫                  0                  ∞                                ⁢                                                                                                    (                                                                              x                            ⁡                                                          (                              t                              )                                                                                -                                                      μ                            x                                                                          )                                            ⁢                                              (                                                                              s                            ⁡                                                          (                                                                                                α                                  k                                                                ⁢                                t                                                            )                                                                                -                                                      μ                                                          s                              ,                              k                                                                                                      )                                                                                                            σ                        x                                            ⁡                                              (                                                                              σ                            s                                                    /                                                                                    α                              k                                                                                                      )                                                                              ⁢                                                                          ⁢                  dt                                                                    ,                  1          <          k          <          K                ,                            (        2        )            where β and σ represent the mean and L2 norm of each signal,
                                          μ            x                    =                                    lim                              T                →                ∞                                      ⁢                                          1                T                            ⁢                                                ∫                  0                  T                                ⁢                                                      x                    ⁡                                          (                      t                      )                                                        ⁢                                                                          ⁢                  dt                                                                    ⁢                                  ⁢                                  ⁢                                  ⁢                              σ            x                    =                                                                      ∫                  0                  ∞                                ⁢                                                                                                                                                    x                          ⁡                                                      (                            t                            )                                                                          -                                                  μ                          x                                                                                                            2                                    ⁢                                                                          ⁢                  dt                                                      .                                              (        3        )            
The correlation coefficient between the optimally scaled baseline s(α t) and observed data x(t) can then be expressed as
                                          ϕ            xs                    =                                    max              k                        ⁢                                          ∫                0                ∞                            ⁢                                                                                          (                                                                        x                          ⁡                                                      (                            t                            )                                                                          -                                                  μ                          x                                                                    )                                        ⁢                                          (                                                                        s                          ⁡                                                      (                                                                                          α                                k                                                            ⁢                              t                                                        )                                                                          -                                                  μ                                                      s                            ,                            k                                                                                              )                                                                                                  σ                      x                                        ⁡                                          (                                                                        σ                          s                                                /                                                                              α                            k                                                                                              )                                                                      ⁢                dt                                                    ,                  1          <          k          <          K                ,                            (        4        )            where value of φxs has a range −1≤φxs≤1. As K→∞, φxs becomes scale-invariant. So if x(t) and s(t) are scaled replicas of one another (i.e., x(t)=s(α t)), then the value of φxs is 1. Conversely, if x(t)=−s(α t), then the value of φxs is −1. When there is no scale relationship between the two signals, φxs=0. For finite values of K and situations in which x(t)≈s(α t), φxs describes the degree of linear correlation between x(t) and the optimally scaled baseline s(α t).
Since OSS is an optimal technique, assuming the uniform scaling model described in (1), system 110 uses it as a baseline for comparing the techniques described herein and other techniques. Although effective, OSS is computationally inefficient. For a discrete signal of N samples and a library of K stretched baselines, the OSS correlation coefficient can be computed using a matrix-vector multiplication. The computationally complexity of this operation, in big-O notational, is O(K N). Therefore, for sufficiently large signals and a sufficiently dense library, OSS becomes computationally impractical.
Local Peak Coherence
A second prior art approach to ultrasonic temperature compensation is referred to as the local peak coherence (LPC) technique, which is used to estimate the scale factor from local delay measurements. It has been experimentally shown that diffuse-like signals, cluttered by reflections and multi-path propagation, also exhibit almost perfect signal stretching behavior as temperature varies. Since the diffuse-like condition ensures the measured signal to have a long and continuous duration, the scale factor between the two signals can be estimated from a series of local delay estimates. LPC assumes that, given an observed signal x(t) and uniformly scaled replica x(t)=s(α t), the uniform scaling effect can be approximated as a delay in a small region around t=t0,x(t)=s(αt)≈s(t−(1+α)t0) as t→t0.  (5)
To estimate these delays, a small portion of each signal is windowed around time t=t0 and the delay between each windowed signal is computed as the argument that maximizes the standard cross-correlation (coherence) function,
                                          d            w                    =                      arg            ⁢                                          max                t                            ⁢                                                ∫                  0                  ∞                                ⁢                                                                            x                      w                                        ⁡                                          (                      τ                      )                                                        ⁢                                                            s                      w                                        ⁡                                          (                                              t                        +                        τ                                            )                                                        ⁢                                                                          ⁢                  dτ                                                                    ,                  1          <          w          <          W                ,                            (        6        )            
where dw is the delay estimate for window w, sw(t) and xw(t) are windowed baseline and observed signals, and W is the number of window positions. As the window moves across each signal, the delay value increases linearly according to the scale factor. The scale parameter is then computed by performing a regression analysis on the estimated delays. Using the computed scale factor, the correlation coefficient between the appropriately scaled baseline and measured data is then computed and used as the scale-invariant statistic.
As compared with OSS, LPC can be computed very quickly. The delay estimates may be computed using the fast Fourier transform (FFT) algorithm. Assuming, Nw is the number of discrete samples in the window and there are W different window positions, computing every delay requires a computational complexity of O(W Nw log(Nw)). In general, Nw will be small. So overall, the computational efficiency is linearly dependent on W, and at a minimum, the scale factor can be estimated using W=2. However, this efficiency comes at the cost of robustness. The technique approximates uniform scaling as a series of delays, which, as shown in equation (5), is true asymptotically as t→t0. In general, this may not always be a good approximation. Also, any effect that does not uniformly scale the signal may disrupt a portion of a delay estimates and may significantly alter the scale factor estimate. For this reason, LPC is not robust and may be adversely affected by many effects, including the introduction of damage.
Therefore there is still a need for methods for compensation for temperature variations in ultrasonic inspection systems that provide high quality results and do so rapidly.